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arXiv 1806.07033math.FA
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Hermitian半正定矩阵平行和的扰动估计

Perturbation estimation for the parallel sum of Hermitian positive semi-definite matrices

Wei Luo, Chuanning Song, Qingxiang Xu

备注：
15 pages, to appear in Linear Multilinear Algebra
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摘要：

设$\mathbb{C}^{n\times n}$是所有$n \times n$复矩阵的集合。对于$\mathbb{C}^{n\times n}$中的任意Hermite半正定矩阵$A$和$B$，构造了它们小于$A+B-A:B$的新的公共上界，其中$(A+B)^†$表示$A+B$的Moore-Penrose逆，$A:B=A(A+B)^†B$是$A$与$B$的并行和.导出了$(A+X):(B+Y)-A:B-X:Y$的一个分解公式，其中$X,Y\in\mathbb{C}^{n\times n}$分别是$A$和$B$的任意Hermite半正定扰动.基于导出的因子分解公式和构造的$X$和$Y$的公共上界，给出了$(A+X):(B+Y)-A:B$的一些新的、尖锐的范数上界.数值例子也说明了所得到的范数上界的尖锐性。

英文摘要：

Let $\mathbb{C}^{n\times n}$ be the set of all $n \times n$ complex matrices. For any Hermitian positive semi-definite matrices $A$ and $B$ in $\mathbb{C}^{n\times n}$, their new common upper bound less than $A+B-A:B$ is constructed, where $(A+B)^†$ denotes the Moore-Penrose inverse of $A+B$, and $A:B=A(A+B)^†B$ is the parallel sum of $A$ and $B$. A factorization formula for $(A+X):(B+Y)-A:B-X:Y$ is derived, where $X,Y\in\mathbb{C}^{n\times n}$ are any Hermitian positive semi-definite perturbations of $A$ and $B$, respectively. Based on the derived factorization formula and the constructed common upper bound of $X$ and $Y$, some new and sharp norm upper bounds of $(A+X):(B+Y)-A:B$ are provided. Numerical examples are also provided to illustrate the sharpness of the obtained norm upper bounds.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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